Physical Chemistry Third Edition

C A Short Table of Integrals 1259

∫∞

0

x^2 e−ax

2

dx

1

4

√

π

a^3

, a> 0

∫∞

0

x^3 e−ax

2

dx

1

2 a^2

, a> 0

∫∞

0

x^2 ne−ax

2

dx

(1)(3)(5)...(2n−1)

2 n+^1 an+^1 /^2

√

π, a> 0

∫∞

0

x^2 n+^1 e−ax

2

dx

n!

2 an+^1

, a> 0

C.3 The Error Function

The error function is defined by

erf(z)

2

√

π

∫z

0

e−t

2

dt

So that

erf(0) 0

erf(1/

√

2) 0. 683 ...

erf(∞) 1

erf(− 1 /

√

2)− 0. 683 ···

erf(−∞)− 1

The following identity is sometimes useful:

∫z

0

t^2 e−at

2

dt

√

π

4 a^3 /^2

erf(

√

az)−

z

2 a

e−az

2

The error function cannot be expressed in closed form (as a formula with a finite number

of terms) for values ofzother than 0 or∞. The following asymptotic formula^2 gives

values of the error function for large arguments:

erf(z) 1 −

e−z

2

√

πz

(

1 +

∑∞

m 1

(−1)m(1)(3)...(2m−1)

(2z^2 )m

)

(^2) M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions with Formulas, Graphs, and
Mathematical Tables, National Bureau of Standards, Washington, DC, 1964, p. 298.